We consider the system AotoYamada_05__Ex5TermProof.

  Alphabet:

    0 : [] --> a 
    add : [a * a] --> a 
    fact : [] --> a -> a 
    mult : [] --> a -> a -> a 
    rec : [a -> a -> a * a] --> a -> a 
    s : [a] --> a 

  Rules:

    add(0, x) => x 
    add(s(x), y) => s(add(x, y)) 
    mult 0 x => 0 
    mult s(x) y => add(mult x y, y) 
    rec(f, x) 0 => x 
    rec(f, x) s(y) => f s(y) (rec(f, x) y) 
    fact => rec(mult, s(0)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

In order to do so, we start by eta-expanding the system, which gives:

  add(0, X) => X 
  add(s(X), Y) => s(add(X, Y)) 
  mult(0, X) => 0 
  mult(s(X), Y) => add(mult(X, Y), Y) 
  rec(F, X, 0) => X 
  rec(F, X, s(Y)) => F s(Y) rec(F, X, Y) 
  fact(X) => rec(/\x./\y.mult(x, y), s(0), X) 

We thus obtain the following dependency pair problem (P_0, R_0, static, formative):

  Dependency Pairs P_0:

    0] add#(s(X), Y) =#> add#(X, Y)   
    1] mult#(s(X), Y) =#> add#(mult(X, Y), Y)   
    2] mult#(s(X), Y) =#> mult#(X, Y)   
    3] rec#(F, X, s(Y)) =#> rec#(F, X, Y)   
    4] fact#(X) =#> rec#(/\x./\y.mult(x, y), s(0), X)   
    5] fact#(X) =#> mult#(Y, Z)   

  Rules R_0:

    add(0, X) => X 
    add(s(X), Y) => s(add(X, Y)) 
    mult(0, X) => 0 
    mult(s(X), Y) => add(mult(X, Y), Y) 
    rec(F, X, 0) => X 
    rec(F, X, s(Y)) => F s(Y) rec(F, X, Y) 
    fact(X) => rec(/\x./\y.mult(x, y), s(0), X) 

Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.

We consider the dependency pair problem (P_0, R_0, static, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0 
    * 1 : 0 
    * 2 : 1, 2 
    * 3 : 3 
    * 4 : 3 
    * 5 : 1, 2 

This graph has the following strongly connected components:

  P_1:

    add#(s(X), Y) =#> add#(X, Y)   

  P_2:

    mult#(s(X), Y) =#> mult#(X, Y)   

  P_3:

    rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_3, R_0, static, formative) is finite.

We consider the dependency pair problem (P_3, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(rec#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(rec#(F, X, s(Y))) = s(Y) |> Y = nu(rec#(F, X, Y)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_3, R_0, static, f) by ({}, R_0, static, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.

We consider the dependency pair problem (P_2, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(mult#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(mult#(s(X), Y)) = s(X) |> X = nu(mult#(X, Y)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by ({}, R_0, static, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.

We consider the dependency pair problem (P_1, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(add#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(add#(s(X), Y)) = s(X) |> X = nu(add#(X, Y)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.